We present efficient implementations of a number of operations for quantum computers. These include controlled phase adjustments of the amplitudes in a superposition, permutations, approximations of transformations and generalizations of the phase adjustments to block matrix transformations. These operations generalize those used in proposed quantum search algorithms.

The position of a point after some rotation about the origin can simply be obtained by multiplying its coordinates with a matrix. One reason for introducing homogeneous coordinates is to be able to describe translation by a matrix so that multiple transformations, whether each is a rotation or a translation, can be concatenated into one described by the product of their respective matrices. How...

We provide a comprehensive analysis of matrix-valued Herglotz functions and illustrate their applications in the spectral theory of self-adjoint Hamiltonian systems including matrix-valued Schrödinger and Dirac-type operators. Special emphasis is devoted to appropriate matrix-valued extensions of the well-known Aronszajn-Donoghue theory concerning support properties of measures in their Nevanli...

Program transformations are widely used in synthesis, optimization, and maintenance of software. Correctness of program transformations depends on preservation of some important properties of the input program. By regarding programs as Kripke structures, many interesting properties of programs can be expressed in temporal logics. In temporal logic, a formula is interpreted on a single program. ...

A numerical method for computing the logarithm of a symmetric positive dee-nite matrix is developed in this paper. It is based on reducing the original matrix to a tridiagonal matrix by orthogonal similarity transformations and applying Pad e approximations to the logarithm of the tridiagonal matrix. Theoretical studies and numerical experiments indicate that the method is quite eecient when th...

Concise variable transformations between the four angles of the CKM matrix in the standard representation advocated by the Particle Data Group and the angles of the unitarity triangles are derived. The behavior of these transformations in various limits is explored. The straightforward extension of this calculation to other representations and more generations is indicated. Typeset using REVTEX

The conditions under which matrix orientifolding and supersymmetry transformations commute are known to be stringent. Here we present the cases possessing four or eight supercharges upon Z3 orbifolding followed by matrix orientifolding. These cases descend from the matrix models with eight plus eight supercharges. There are fifty in total, which we enumerate. e-mail: [email protected] ...

We adapt White’s density matrix renormalisation group (DMRG) to the direct study of critical phenomena. We use the DMRG to generate transformations in the space of coupling constants. We postulate that a study of density matrix eigenvalues leads to a natural identification of renormalised blocks, operators and Hamiltonians. We apply the scheme to the phase transition in the anisotropic spin-1/2...

Some perturbation estimates for the unique positive definite solution of a nonlinear matrix equation connected to the interpolation theory are derived. The considered estimations are modification of some existing one. They are obtained by similar transformations of the matrix coefficients with a positive definite matrix. The theoretical results are illustrated by several numerical examples.

We propose a novel statistical descriptor, Multiple References Histogram Matrix (MRHM), for robust shape retrieval, especially for degraded shape images. For each shape image, MRHM first generates uniform grids and filters noises in each grid by line Hough transformations and curve-fitting transformations. Then MRHM selects a reference for each grid and calculates its local distribution between...